Here’s one we haven’t seen (and which I’ll talk more about later): given any category , the category of “endofunctors” has a monoidal structure given by composition of functors from to itself. This is the one I was thinking of that doesn’t have a symmetry, by the way. A monoid object in this category consists of a functor along with natural transformations and . These turn out to be all sorts of useful in homology theory, and also in theoretical computer science. In fact, the programming language Haskell makes extensive and explicit use of them.

And now for a really interesting class of examples. Let’s say we start with a monoidal category with monoidal structure . We immediately get a monoidal structure on the opposite category. Just define for objects. For morphisms we take and (which are in and , respectively), and define , which is in .

So what’s a monoid object in ? It’s a contravariant functor from to . Equivalently, we can write it as a covariant functor from to . It will be easier to just write down explicitly what this opposite category is.

So we need to take and reverse all the arrows. It’s enough to just reverse the arrows we threw in to generate the category, and their composites will be reversed as well. We’ll also have to dualize the relations we imposed to make everything work out right. So we’ll have an arrow called comultiplication and another arrow called the counit. These we require to satisfy the coassociative condition and the left and right coidentity conditions .

Now a functor from this category to another monoidal category picks out an object and arrows (reusing the names) and satisfying coassociativity and coidentity conditions. We call such an object with extra structure a “comonoid object” in . In we call them “comonoids”. In we call them “corings” (with counit), in we call them “coalgebras” (with counit), and in we call them “comonads”. In general, we call this new model category — the “theory of comonoids”.

[…] object in the category of vector space over . Then a compatible coalgebra structure makes it a comonoid object in the category of algebras over . Or in the other order, we have a monoid object in the category […]

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